Abstract

Finding optimal dosing strategies for treating bacterial infections is extremely difficult, and improving therapy requires costly and time-intensive experiments. To date, an incomplete mechanistic understanding of drug effects has limited our ability to make accurate quantitative predictions of drug-mediated bacterial killing and impeded the rational design of antibiotic treatment strategies. Three poorly understood phenomena complicate predictions of antibiotic activity: post-antibiotic growth suppression, density-dependent antibiotic effects, and persister cell formation. We show that chemical binding kinetics alone are sufficient to explain these three phenomena, using single-cell data and time-kill curves of Escherichia coli and Vibrio cholerae exposed to a variety of antibiotics in combination with a theoretical model that links chemical reaction kinetics to bacterial population biology. Our model reproduces existing observations, has a high predictive power across different experimental setups (R(2) = 0.86), and makes several testable predictions, which we verified in new experiments and by analyzing published data from a clinical trial on tuberculosis therapy. Although a variety of biological mechanisms have previously been invoked to explain post-antibiotic growth suppression, density-dependent antibiotic effects, and especially persister cell formation, our findings reveal that a simple model that considers only binding kinetics provides a parsimonious and unifying explanation for these three complex, phenotypically distinct behaviours. Current antibiotic and other chemotherapeutic regimens are often based on trial and error or expert opinion. Our "chemical reaction kinetics"-based approach may inform new strategies, which are based on rational design.

Time-lapse microscopy of single-E. coli cells observed (imaging rate: 1/5 minutes) during and after a 16 h exposure to two concentrations: 22 cells at 6.25 mg/L (0.4 MIC) (A) and 27 cells at 12.5 mg/L (0.8 MIC) (B) of tetracycline, corresponding to 0.4 and 0.8 MIC, respectively. The solid black line indicates the mean growth rate of the observed bacterial population, the dotted blue lines indicate the minimal and maximal observed growth rate. The model used to fit the data is described in and methods. Kinetic parameters are derived from literature cited in .

These graphs show the relative bacterial population size over 12h with different tetracycline concentrations as compared to growth in absence of antibiotics. (A) E. coli were grown in microtiter plates with different initial densities and exposed to different tetracycline concentrations (the MIC is 2 mg/L); the graph shows relative population size in presence vs absence of antibiotic. The average of three independent experiments is shown. (B) Model prediction using parameters from single cells; (C) Predictive power of model. The y-axis shows the average bacterial density over 12 h as observed by turbidity measurements in microtiter plates. Bacterial cultures with different initial densities were exposed to varying concentrations of tetracycline (Fig. 2A). The x-axis shows the theoretical prediction derived from our model parameterized with known binding data and the parameter estimates from fitting the model to single-cell data (). (D) Correlation between strength of inoculum effect (efficacy loss/ log10 bacteria) and drug-target affinity (half-maximal target binding) in E. coli. For details see and methods, “Quantification of inoculum effect”. The experimental setup was the same as in (A). (E) Same as in (D) for Vibrio cholerae. (Amp: Ampicillin; Gen: Gentamicin; Nal: Nalidixic Acid; Str: Streptomycin; Tet: Tetracycline; Cip: Ciprofloxacin; Pen: Penicillin; Rif: Rifampicin).

Model of persistence as tolerant subpopulations with phenotypic switch compared to alternative model

(A) This graph shows numerical simulations of the mathematical model proposed by Balaban et al. (). In brief, this model assumes that there is a majority population of susceptible (‘normal’) bacteria n, and a subpopulation of persistent bacteria, p. According to the data in () (represented by red line), the majority population declines quickly with a rate μn of ∼0.4 orders of magnitude per hour (μn=-1.84 h-1) at 100 mg/L ampicillin, the persister population is not affected by the antibiotic (μp=0), and bacteria switch from a persistent to a normal state with a rate b=0.07 h-1. This can be described with the following mathematical model as presented in (): (10) The black line represents an antibiotic activity that is 50% lower than 100 mg/L; the green line indicates an antibiotic activity that is twice higher. (B), (C) Distribution of susceptibility in bacterial population with different explanations for persistence. On the x-axis, the antibiotic susceptibility of individual cells is given relative to the mean of the non-persister population, the y-axis shows the number of bacteria with that specific susceptibility (total population size 107). (B) ‘Classical model’ of persistence with a majority non-persister population and a minority persister population. The relative proportion of persisters under this assumption is usually assumed to be several orders of magnitude lower than the majority population. For illustration purposes only, we adopt a persister frequency of 10-2 (the real frequency is much lower and would not be visible in this figure).

A normal distribution of target molecules can lead to multiphasic bacterial killing

(A) Histogram of intracellular ribosomal concentration as experimentally determined in E. coli. Raw data were obtained from (). Normal distribution was not rejected by Shapiro-Test, the red line shows a normal distribution with mean and standard deviation obtained from the data. (B) Predicted killing over time for bacterial population measured in (A). Kinetic rates of streptomycin binding to ribosomes were taken from the literature (). Time of killing was calculated using . The colors indicate antibiotic concentrations relative to the MIC of E. coli MG1655. (C) Theoretical explanation of multiphasic kill curves. The dotted lines show the calculated equilibrium number of bound ribosomes for three different streptomycin concentrations. The solid lines show the time until a specific fraction of the target molecules is bound (, methods). Black: assumed killing threshold with a mean of 60% and variance of 10%. (D) and (E) These graphs show time-kill curves (E) resulting from a bimodal distribution of susceptibility ((D), compare to ). The MIC was calculated as the drug concentration that achieves binding of 99% of the cells at equilibrium (). The vertical lines in (D) indicate the equilibria at given multiples of the MIC (green= 1×MIC, yellow= 4× MIC, orange= 8×MIC, red= 32×MIC). The same colors are used in the resulting calculated time-kill curves of a simulated population of 105 bacteria in (E). Kinetic rates are the same as in (A)- (C). The main “peak” of the bimodal distribution in (D) corresponds to the normal distribution in .

(A) Experimental time-kill curves of E. coli with different ciprofloxacin concentrations. The MIC is 8mg/L (). The experiment was performed in biologically independent triplicates (B) Numerical simulations of a model combining bacteriostatic and bacteriocidal effects () parameterized with in vitro kinetic parameters for ciprofloxacin binding. In the absence of known distributions of functional gyrase tetramers, we assume that the mean of this distribution is the average of the numbers given in the literature as cited in . We assume a normal distribution in absence of more detailed information (, ) and that the standard deviation is the same as published for ribosomes.

(A)-(C): Colony-forming unit counts of M. tuberculosis in sputum during treatment as reported in (). Patients with previously untreated, smear-positive pulmonary tuberculosis due to drug-sensitive strains received 14 days initial monotherapy and then a standard multi-drug regimen. Sputum was collected overnight on two consecutive nights before treatment was started and then after 2, 4, 6, 8, 10, 12, and 14 days of the allocated regimen. Patients received (A) 5mg rifampicin/kg bodyweight, (B) 10mg rifampicin/kg bodyweight (rifampicin dosage in standard regimen), (C) 20mg rifampicin/kg bodyweight. (D) Calculated time-kill curves within patients (simulated population of 104 bacteria, ). The effective drug concentration in extracellular lining fluid was obtained from (). The MIC for rifampicin in drug-susceptible strains was assumed to be 0.1 mg/L (). The available drug concentration was assumed to follow a normal distribution with a 22.5% standard deviation ().

(A) INH forms an NAD-adduct (red spheres) with a rate which depends on the enzyme KatG and O2 saturation. INH-NAD then reacts with its target InhA (blue crescents). (B) Time-kill curves resulting from calculating the expected time until 60% (with 10% standard deviation) of the target molecules are bound (simulated population of 106 bacteria, , methods). Black indicates the same intracellular oxygen content as in the cell-free experimental setup in (), red 50% and green 25% O2 saturation. (C) Time-kill curves for variance in intracellularly available KatG by 30%, simulated population of 104 bacteria. All other parameters from (). Black indicates the MIC for M. tuberculosis (), red 2× MIC and green 3× MIC.

(A), (B) Model overview. A bacterial cell with target molecules, T (in this case ribosomes, blue crescents) and antibiotic molecules A (red circles) are shown. Antibiotics must diffuse through the bacterial cell envelope with a diffusion constant, p, in order to bind their targets with an association rate kf to form a complex AT, which may dissociate with a rate kr (0 for irreversible reactions). (C) Different mechanisms can lead to bacterial killing. Bacterial cells are shown as black oval, target molecules, are shown as blue crescents, and antibiotic molecules as red circles. (D) Chemical reactions of bacterial cells. For an explanation of the parameters see main text. The color of the reaction arrow is used to highlight the corresponding equations in and methods.